The Probably Approximately Correct (PAC) learning model is what formally justifies the tendency of neural networks to "learn" from data (see Wikipedia).

While the PAC model does not depend on the probability distribution which generates training and test data, it does assume that they are *the same*. So by "adversarially" choosing test data, the researchers are breaking this important assumption. Therefore it is in some ways not surprising that neural networks have this vulnerability. It shouldn't be an issue in real life, assuming that the training data and the testing data really do come from the same probability distribution.

That said, this shows why you wouldn't want to use neural networks for, say, cryptography.

I agree; I just read the original recently. Of course the invisible man was an albino, so he was already pigmentless (or so the story goes) except for blood etc. These pigments he somehow replaced. A lot of the scientific explanation has to do with changing the refractive indices of his tissues to make them match those of air though. It seems the mouse still has some problems with refraction.

As Nick Boston pointed out (http://www.nickbostrom.com/extraterrestrial.pdf)

this is the worst news the human race has ever received.

The idea is that the Fermi Paradox must be the result of a Great Filter which stymies the creation of long lived intelligent races. The easier it is for life to evolve, the more likely it is that the Filter lies ahead of us, rather than behind.

I'm not a programmer, I'm a mathematician, but I notice the same thing in my field.

To those who say there is not a tendency toward weirdness in mathematical disciplines, I suggest the following experiment. First go to the weekly math colloquium at a local research university. Then, go to the weekly philosophy colloquium and see if you can discern a difference in the people who come. I believe you will almost certainly find that the mathematicians are less attractive and charismatic. You could argue that philosophy simply selects for attractiveness and charisma, but I believe you will have similar findings if many different subjects are substituted for phil.

To those who say that the strangeness of programmers is somehow reducible to various qualities of "geeks", this is clearly begging the question, as any good geek should know. The topic for this thread is very similar to asking "why are geeks the way they are?" but phrased differently.

I have spent large amounts of time wondering why mathematicians are weird, ugly, uncharismatic and so forth. My answer is that they live largely in their own imaginations, and spend correspondingly less time in the "real world." Therefore, not surprisingly, their real world appearance, manners etc gives evidence of a lack of attention. Conversely people in other fields are not selected for an ability to concentrate deeply, spend more time in the here and now, and reap consequent benefits in hygiene, social skills, etc.

I assume you are interested only in original documents, and not summaries or expositions. Unfortunately the technical and specialized nature of modern science is likely to make "accessible science" and "original science" mutually exclusive.

I'm a mathematics professor. I first became interested in math after reading Excursions in Number Theory during my junior year of college. It is a wonderful introduction to the power of proof, and requires no more background than simple arithmetic.
I second someone's earlier suggestion of "e" by Eli Maor. That is truly an outstanding book.
Unfortunately what's good for a high school student may not be good for the rest of us. I found Zero: The History of a Dangerous Idea to be completely vapid, but it is full of intrigue and controversy, and is not difficult.
A genuinely good book is John Derbyshire's history of algebra, though it is challenging in places.
It is worth remembering that Ramanujan (according to myth) was strongly motivated by the book "A synopsis of elementary results in pure and applied mathematics", written by George S. Carr. It shows that a book of facts (such as Excursions) may be as good or better than a popular or historical book.
I will go out on a limb and make a strange suggestion: Tractatus Logico Philosophicus, by Wittgenstein is completely fascinating, and no less comprehensible to a high school student than to the rest of us. It may create an appreciation for the mysterious and profound aspects of mathematics that could be powerfully motivational.

I'm a recent graduate with a PhD in mathematical logic; I can totally relate to this problem of having a non-standard background. Before Grad School I went to a liberal arts college, where my math major consisted of something like 9 courses. When my advanced studies began I felt totally lost.
But you have to ask yourself: Do I really have the time and energy to commit to a high level exposition of physics at this point? The answer to this question will depend on whether you intend to specialize in PDE's. If the answer is no, then I believe you should buy some good popular explication of the topics you're covering.
Unfortunately, I do not know what such a book would be, but having read extremely good expositions of several high level mathematical concepts (Prime Obsession; Unknown Quantity; "e", the Story of a Number, and Incompleteness) I have some confidence that a book of similar quality may exist in this area.
Of course, if you are intending to specialize in PDE's, it will be worth the time and effort you will need to invest in reading a serious text. Even so, keep in mind that you can never learn everything, even about a small subdiscipline. My advice is to find a particular area of PDE's, become an expert on that, and get out of graduate school as soon as possible.

sevenfactorial writes: "I'm a soon-to-be recent graduate from a PhD program in mathematics. Academia is starting to seem less interesting than in did when I began my program (seven years ago), and I'm interested in finding work in the software industry. The problem is that nothing seems particularly well-suited to my background — software development jobs usually require only a BS, whereas senior developer positions require technical knowledge which I don't have. I have great technical fluency, and I can solve problems well, but I don't have the particular industry skills to start at a level that seems commensurate with my education. Does anyone have any experience with transitioning from an abstruse academic speciality to interesting work in IT? Any anecdotes or advice would be appreciated."